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Properties

Label 4-1568e2-1.1-c1e2-0-29
Degree $4$
Conductor $2458624$
Sign $1$
Analytic cond. $156.763$
Root an. cond. $3.53843$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 3·9-s − 4·11-s − 8·13-s + 2·17-s − 6·19-s + 8·23-s + 5·25-s − 10·27-s + 4·29-s − 4·31-s + 8·33-s − 10·37-s + 16·39-s − 20·41-s − 8·43-s + 4·47-s − 4·51-s + 2·53-s + 12·57-s + 10·59-s + 8·61-s − 8·67-s − 16·69-s + 6·73-s − 10·75-s − 16·79-s + ⋯
L(s)  = 1  − 1.15·3-s + 9-s − 1.20·11-s − 2.21·13-s + 0.485·17-s − 1.37·19-s + 1.66·23-s + 25-s − 1.92·27-s + 0.742·29-s − 0.718·31-s + 1.39·33-s − 1.64·37-s + 2.56·39-s − 3.12·41-s − 1.21·43-s + 0.583·47-s − 0.560·51-s + 0.274·53-s + 1.58·57-s + 1.30·59-s + 1.02·61-s − 0.977·67-s − 1.92·69-s + 0.702·73-s − 1.15·75-s − 1.80·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2458624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2458624 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2458624\)    =    \(2^{10} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(156.763\)
Root analytic conductor: \(3.53843\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 2458624,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$Isogeny Class over $\mathbf{F}_p$
bad2 \( 1 \)
7 \( 1 \)
good3$C_2^2$ \( 1 + 2 T + T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.3.c_b
5$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.5.a_af
11$C_2^2$ \( 1 + 4 T + 5 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.11.e_f
13$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.13.i_bq
17$C_2^2$ \( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.17.ac_an
19$C_2^2$ \( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.19.g_r
23$C_2^2$ \( 1 - 8 T + 41 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.23.ai_bp
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.29.ae_ck
31$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.31.e_ap
37$C_2$ \( ( 1 - T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.37.k_cl
41$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.41.u_ha
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.43.i_dy
47$C_2^2$ \( 1 - 4 T - 31 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.47.ae_abf
53$C_2^2$ \( 1 - 2 T - 49 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.53.ac_abx
59$C_2^2$ \( 1 - 10 T + 41 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.59.ak_bp
61$C_2^2$ \( 1 - 8 T + 3 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.61.ai_d
67$C_2^2$ \( 1 + 8 T - 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.67.i_ad
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2^2$ \( 1 - 6 T - 37 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.73.ag_abl
79$C_2^2$ \( 1 + 16 T + 177 T^{2} + 16 p T^{3} + p^{2} T^{4} \) 2.79.q_gv
83$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.83.e_go
89$C_2^2$ \( 1 + 18 T + 235 T^{2} + 18 p T^{3} + p^{2} T^{4} \) 2.89.s_jb
97$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.97.e_hq
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.418564132729568135703695940938, −8.563899505952502567050134363861, −8.562303252400132648517159579864, −8.138431466689679123969702062916, −7.29143281399030469363913597249, −7.08226402182520783851452356127, −7.00764085731017573488002496055, −6.57774818461754523116741836472, −5.67261209230861090809381373962, −5.53398717231615884105226253317, −4.96609649475725695311260229075, −4.94457999851247848760270108663, −4.45274012504506437945278018808, −3.63832345110949639900419483165, −3.16724902907722981967658499841, −2.52925504706822582949965041094, −2.05135784993971276255827462844, −1.31463107457763231150686650869, 0, 0, 1.31463107457763231150686650869, 2.05135784993971276255827462844, 2.52925504706822582949965041094, 3.16724902907722981967658499841, 3.63832345110949639900419483165, 4.45274012504506437945278018808, 4.94457999851247848760270108663, 4.96609649475725695311260229075, 5.53398717231615884105226253317, 5.67261209230861090809381373962, 6.57774818461754523116741836472, 7.00764085731017573488002496055, 7.08226402182520783851452356127, 7.29143281399030469363913597249, 8.138431466689679123969702062916, 8.562303252400132648517159579864, 8.563899505952502567050134363861, 9.418564132729568135703695940938

Graph of the $Z$-function along the critical line